Theory of Probability &Its Applications, Volume 65, Issue 1, Page 49-61, January 2020.
We consider the problem of constructing guarantee procedures of statistical inference with fixed minimal observation number $n^*$ for discrimination of two hypotheses $H_0\colon\theta\in[\theta_1,\theta_2]$ and $H_1\colon\theta\notin[\theta_1,\theta_2]$ with a one-dimensional parameter $\theta$ under the so-called $d$-posterior approach. Here, constraints are placed on the conditional probabilities for the validity of one or another hypothesis under the condition that this hypothesis is rejected. We give an asymptotic formula for $n^*$ in a scheme with severe (tending to zero) constraints on these conditional probabilities of hypotheses. Earlier, Volodin and Novikov found a similar formula for discrimination of one-sided hypotheses. In the present paper, the proof of the asymptotic formula is carried out under weaker constraints on the probability model. The accuracy of our formula is illustrated numerically for some probability models.

中文翻译：

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$ d $保证的双向假设的最小辨别的最小充分观察数的渐近性

概率论及其应用，第65卷，第1期，第49-61页，2020年1月。
我们考虑构造具有固定最小观测值$ n ^ * $的统计推断的保证程序，以区别两个假设$ H_0 \ colon \ theta \ in [\ theta_1，\ theta_2] $和$ H_1 \ colon \ theta \ notin [\ theta_1，\ theta_2] $在所谓的$ d $-后验方法下具有一维参数$ \ theta $。在此，在一个假设被拒绝的条件下，对一个或另一个假设的有效性的条件概率施加了约束。在这些假设的条件概率具有严格（趋于零）约束的方案中，我们给出了$ n ^ * $的渐近公式。早些时候，Volodin和Novikov发现了用于区分单方面假设的类似公式。在本文中，渐近公式的证明是在概率模型的较弱约束条件下进行的。对于某些概率模型，我们用数字说明了公式的准确性。